14.04 is shipping a too old version of Qt. QtWebEngine is a pretty new module so for Ubuntu to get it into its official repository means that it could not be in the current either. A quick check to packages.ubunto.com confirmed it.

Indeed, WebKit has been deprecated and won't, therefore, be shipping with the forthcoming Qt 5.6. However, you can still build from source (or, at least, you should still be able to build it from source).

As a general rule, new projects ought to use WebEngine, but because there isn't yet a 1:1 correspondance between WebEngine and WebKit (is there ever going to be?!), you might have no choice but have to use WebKit for the time being.

(I am in that boat too, i.e. WebEngine is currently missing too many things to make it possible for me to drop WebKit -- I tried! --. Yet, I can't see myself building WebKit from source, so I am not going to upgrade to Qt 5.6. In the end, I really wish Qt would release binaries for WebKit. Not as part of Qt 5.6 since it's going to be an LTS, but at least on the side so many people (I am sure) don't feel like their only options are either to build WebKit from source (!!) or stick to Qt 5.5.)

QtWebKit was not available on Android either, it's one of the platform that, with iOS, doesn't allow third party web engines. You can use the QtWebView module for these platforms. That will provide you a native web view to use.

Welp, I'm VERY close at this point thanks to some helpful information on the ODROID forums (I am building using a Pi2 and an ODROID C1). At this point, I'm stumbling on Chromium, which the current set up seems to be setting it for a Linux desktop build running gyp_run.pro. The problem is that this causes the following error:

gyp name 'target_arch' is not defined while evaluating condition 'use_qt==1 and target_arch=="ia32"' in /home/flame/builds/qt5/qtwebengine/src/3rdparty/chromium/base/base.gyp

I'm still trying to solve the KHR issue on the Pi (despite already updating the header), but for now, the chromium situation is what has me in a gridlock. I can override the desktop build and instead recommend a cross compile in reference to itself, but I'm unsure as to how well this will go.

I am making documentation of what is going on so far so that I can relay an easier to follow guide for people who also are trying ot compile natively for the Pi.